Solicitor General Robert Buckland told BBC News at Ten: "There's an expectation that a discussion will yield some fruit, and I'm not saying it won't and it could very well end up with a further amendment in the Lords."

After Mrs May talked round potential rebels, MPs voted by 324 to 298 to reject a House of Lords amendment that would have given Parliament the power to tell the PM to go back and renegotiate the Brexit deal she secures from Brussels.

Dominic Grieve withdrew his own amendment, which would have given MPs powers to dictate what the Government should do if no acceptable agreement is reached by February 2019.

He said Mrs May promised to table amendments in the House of Lords that will address their concerns.

The senior Remainer said he hoped a compromise would be found and warned "this isn't the end of the matter" if that did not happen.

He told BBC Two's Newsnight: "If it doesn't happen then obviously this isn't the end of the matter because ultimately it is very likely that this amendment will come back."

Mr Grieve said no government would survive if it tried to dispense with Parliament's input.

Conservative MP Dominic Grieve says the government has to "negotiate and compromise" and if it doesn't "it's very likely this amendment will come back [to the Commons]" #newsnightpic.twitter.com/3caPtouaSc

Mrs May will be hoping to avoid defeat in a fresh round of Brexit votes on the EU (Withdrawal) Bill just hours after seeing off the threatened rebellion.

She is expected to come through the latest round of votes unscathed, with a potentially explosive clash over the customs union on Wednesday already defused by a compromise amendment.

Labour's leadership, meanwhile, will be on alert for a potential rebellion after ordering its MPs not to back proposals for a Norway-style agreement with the European Union.

The party has put forward an alternative option to the one backed by peers, which calls for the Government to prioritise being part of the European Economic Area, but some of its MPs have said it does not go far enough.

Mrs May met pro-EU Tories in her private room in the Commons moments before a crucial vote to hear their demands for a truly meaningful vote on the final exit deal.

Potential rebels fell into line after Mr Buckland said ministers were ready to "engage positively" with their concerns before the Bill returns to the Upper House next Monday.

Remainer Stephen Hammond said a group of potential rebels - believed to number 15 to 20 - received assurances from the PM moments before the key vote.

Leading pro-EU Conservative Sarah Wollaston said the "promised further amendment" in the Lords must "closely reflect" the withdrawn proposal.

Earlier, Mrs May was hit by the resignation of justice minister Phillip Lee, who quit the Government live on stage during a speech in London in order to be able to back Mr Grieve's amendment.

A Department for Exiting the European Union spokesman said: "The Brexit Secretary has set out three tests that any new amendment has to meet - not undermining the negotiations, not changing the constitutional role of Parliament and Government in negotiating international treaties, and respecting the referendum result.

"We have not, and will not, agree to the House of Commons binding the Government's hands in the negotiations."

Shadow Brexit secretary Sir Keir Starmer said: "We will wait and see the details of this concession and will hold ministers to account to ensure it lives up to the promises they have made to Parliament."

Liberal Democrat Brexit spokesman Tom Brake said: "As has become a tradition in Brexit negotiations, the Tories have been forced to cobble together a compromise.

"Time will tell as to whether this is just another attempt to buy off the rebels or a real attempt at consensus. But if we face the prospect of a 'meaningless process' rather than a 'meaningful vote', Parliament will be enraged."

Ukip leader Gerard Batten said: "The only 'meaningful vote' was the verdict of the people in referendum of June 23 2016."